Cheese

AtCoder

IDagc056_e

Time5000ms

Memory256MB

Difficulty—

There is a circumference of length $N$. On this circumference, a point whose distance from a certain fixed point, measured in the clockwise direction, is $x$ has the coordinate $x$. For each integer $i$ ($0 \leq i \leq N-1$), there is a mouse at coordinate $i+0.5$. Snuke will throw a piece of cheese $N-1$ times. More specifically, he repeats the following $N-1$ times. * Choose an integer $y$ ($0 \leq y \leq N-1$) randomly, where $y=i$ is chosen with probability $A_i$ percent. This choice is made independently each time. * Then, throw cheese from the coordinate $y$. The cheese travels in the clockwise direction on the circumference. When the cheese meets a mouse, the following happens. * If the mouse has not eaten cheese: * It eats the cheese with probability $1/2$. The cheese disappears. * It does nothing with probability $1/2$. * If the mouse has already eaten cheese: * It does nothing. * The cheese keeps traveling on the circumference until eaten by some mouse. After $N-1$ throws of cheese, there will be exactly one mouse that has never eaten cheese. For each $k=0,1,\cdots,N-1$, find the probability that the mouse at coordinate $k+0.5$ ends up not eating cheese, modulo $998244353$. How to represent a probability modulo $998244353$We can prove that each probability in question is always a rational number. Additionally, under the Constraints of this problem, when that number is represented as an irreducible fraction $\frac{P}{Q}$, we can also prove that $Q \neq 0 \pmod{998244353}$. Thus, there is a unique integer $R$ such that $R \times Q \equiv P \pmod{998244353}, 0 \leq R < 998244353$. Answer with this $R$. ## Constraints * $1 \leq N \leq 40$ * $0 \leq A_i$ * $\sum_{0 \leq i \leq N-1} A_i = 100$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $A_0$ $A_1$ $\cdots$ $A_{N-1}$ [samples]

Samples

Input #1

2
0 100

Output #1

665496236 332748118

$y=1$ is always chosen. When throwing cheese from this point, the following happens.

*   With probability $1/2$, the mouse at coordinate $1.5$ eats it.
*   With probability $1/4$, the mouse at coordinate $0.5$ eats it.
*   With probability $1/8$, the mouse at coordinate $1.5$ eats it.
*   With probability $1/16$, the mouse at coordinate $0.5$ eats it.
*   $\vdots$

In the end, the mice at coordinates $0.5$ and $1.5$ eat this cheese with probabilities $1/3$ and $2/3$, respectively.
Therefore, they end up not eating cheese with probabilities $2/3$ and $1/3$, respectively.

Input #2

2
50 50

Output #2

499122177 499122177

Input #3

10
1 2 3 4 5 6 7 8 9 55

Output #3

333507001 91405664 419217284 757959653 974851434 806873643 112668439 378659755 979030842 137048051

API Response (JSON)

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  "problem": {
    "name": "Cheese",
    "description": {
      "content": "There is a circumference of length $N$. On this circumference, a point whose distance from a certain fixed point, measured in the clockwise direction, is $x$ has the coordinate $x$. For each integer $",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 5000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc056_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a circumference of length $N$. On this circumference, a point whose distance from a certain fixed point, measured in the clockwise direction, is $x$ has the coordinate $x$.\nFor each integer $...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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